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Messy nice stuff & Nice messy stuff. Omri Barak. Collaborators: Larry Abbott David Sussillo Misha Tsodyks. Sloan-Swartz July 12, 2011. Neural representation. Representation of task parameters by neural population. We know that large populations of neurons are involved. - PowerPoint PPT Presentation
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Omri Barak
Collaborators:Larry AbbottDavid SussilloMisha Tsodyks
Sloan-SwartzJuly 12, 2011
Messy nice stuff&
Nice messy stuff
Neural representation
• Representation of task parameters by neural population.
• We know that large populations of neurons are involved.
• Yet we look for and are inspired by impressive single neurons.
• Case study: Delayed vibrotactile discrimination (from Ranulfo Romo’s lab)
f1 f2
time (sec)
Romo & Salinas, Nat Neurosci Rev, 2003
f1 f2
f1>f2?Y N
time (sec)
Romo & Salinas, Nat Neurosci Rev, 2003
Romo task
• Encoding of analog variable• Memory of analog variable• Arithmetic operation “f1-f2”
Romo, Brody, Hernandez, Lemus. Nature 1999Machens, Romo, Brody. Science 2005
• Striking tuning properties• Lead to “simple / low dimensional” models• “Typical” neurons are used to define
model populations.
Existing models
Machens et al. 2005
Miller et al. 2006Barak et al. 2010
Miller et al. 2003
Not shown: VergutsDeco
Singh and Eliasmith 2006
But… Are all cells that good?
Barak et al. 2010
Brody et al. 2003
35% of the neurons flip their sign
-6 -4 -2 0 2 4 6 8-3
-2
-1
0
1
2
3
Stimulus tuning
Del
ay tu
ning
Jun et al. 2010
prestim -1.5 0 0.5 1.5 2.5 3.5 40
20
40
Time (sec)
10 Hz22 Hz34 Hz
Echo state network
Jaeger 2001Maass et al 2002Buonomano and Merzenich 1995
Echo state network
N
ii
Outi
FBi
N
jjiji
i
rwz
zwrJgxdtdx
1
1
x
r
-4 -2 0 2 4-1
-0.5
0
0.5
1
-2 0 2 40
0.5
1
1.5
2
x
+ Noise
N = 1000 / 2000K = 100 (sparseness)g = 1.5
Implementing the Romo task
f1 f2
f
r
Sussillo and Abbott 2009Jaeger and Haas 2004
Input(f1,f2)
Output
Input(f1,f2)
Output
Unit activity
It works, but…
• How does it work?– After the training, we have a network that is
almost a black box.• Relation to experimental data.
Hypothesis
• Consider the state of the network in 1000-D as the trial evolves
00.5
1 0
0.5
10
0.5
1f1
f2
time (sec)
Hypothesis
• Focus only at the end of the 2nd stimulus.• For each (f1,f2) pair, there is a point in
1000-D space.
0.5 1 1.5 20.5
1
1.5
2
0.51
1.52 0.5
1
1.5
2
-1
0
1
0.51
1.52 0.5
1
1.5
2-2
0
2
Hypothesis
• Focus only at the end of the 2nd stimulus.• For each (f1,f2) pair, there is a point in
1000-D space.• So there is a 2D manifold in the 1000-D
space.• Can the dynamics (after learning) draw a
line through this manifold?
0.5 1 1.5 20.5
1
1.5
2
0.51
1.52 0.5
1
1.5
2
-1
0
1
0.51
1.52 0.5
1
1.5
2-2
0
2
Dynamics or just fancy readout?
0 20 40 60 80 100 1200
0.5
1
1.5
2
2.5
3
3.5
4
Yes - YesYes - NoNo - NoFixed - YesFixed - No D
istance in state space
• The two responses are different in network activity, not just through the particular readout we chose.
Saddle point
Searching for a saddle in 1000D
N
ii
Outi
FBi
N
jjiji
i
rwz
zwrJgxdtdx
1
1
ii
FBi
N
jjijii
xfxF
zwrJgxxf
2
1
)()(
)(Vector function:
Scalar function:
Searching for a saddle in 1000D
1
Num
ber o
f uns
tabl
e ei
genv
alue
s
Distance along trajectory
Num
ber o
f un
stab
le e
igen
valu
es
Norm of fixed point
Saddle point
Saddle point
Slightly more realistic
• Positive firing rates• Avoid fixed point between trials. • Introduce reset signal.• Chaotic activity in delay period
= 0
It works
Nice persistent neurons
0 20 40 60 800
0.5
1
1.5
0 20 40 60 800
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
Act
ivity
a1-a2 plane
Romo and Salinas 2003
-2 0 2-2
0
2
-2 0 2-2
0
2
-2 0 2-2
0
2
-2 0 2-2
0
2
f1 tuning
f 2 tu
ning
Problems / predictions
• Reset signal• Generalization
Reset
There is a reset (Barak et al 2010, Churchland et al)
There is no reset, and performance shows it (Buonomano et al 2007)
Time (sec)
Cor
rela
tion
Correlation between trials with different frequencies
Generalization
• Interpolation vs. Extrapolation
f1
f 2
Generalization
• Interpolation vs. Extrapolation
f1
f 2
Generalization
• Interpolation vs. Extrapolation
f1
f 2
Extrapolation
Delosh et al 1997
Conclusions
• Response properties of individual neurons can be misleading.
• An echo state network can solve decision making tasks.
• Dynamical systems analysis can reveal function of echo state networks.
• Need to find a middle ground.